Thesis--Auburn University, 2009. / Abstract. Vita. Includes bibliographical references (p. 41-48).

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 104-107).

Thesis (Ph. D.)--University of Chicago, Dept. of Statistics, August 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph.D.)--Mississippi State University. Department of Chemistry. / Title from title screen. Includes bibliographical references.

Thesis (Ph. D.)--University of Iowa, 1969. / Vita. Includes analyses of selected works by Bartok, Schönberg, Webern, and Scriabin. Photocopy of typescript. Ann Arbor, Mich. : University Microfilms International, 1981. -- 21 cm. Includes bibliographical references (leaves 141-143).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 99-102) and index.

Thesis (doctoral)--Technische Universiteit Eindhoven, 2000. / Includes summary in Dutch. Vita. Includes bibliographical references (p. [105]-110) and index.

In this dissertation, we study the following two topics, i.e., the union closed set conjecture and the maximum edges cut in connected digraphs. The union-closed-set-conjecture-topic goes as follows. A finite family of finite sets is {\it union closed} if it contains the union of any two sets in it. Let $X_{\mathcal{F}}=\cup_{F\in\mathcal{F}}F$. A union closed family of sets is {\it separating} if for any two distinct elements in $\mathcal{F}$, there is a set in $\mathcal{F}$ containing one of them, but not the other and there does not exist an element which is contained in every set of it. Note that any union closed family $\mathcal{F}$ is a poset with set inclusion as the partial order relation. A separating union closed family $\mathcal{F}$ is {\it irreducible} ({\it normalized}) if $|X_{\mathcal{F}}|$ is the minimum (maximum, resp.) with respect to the poset structure of $\mathcal{F}$. In the part of dissertation related to this topic, we develop algorithms to transfer any given separating union closed family to a/an normalized/irreducible family without changing its poset structure. We also study properties of these two extremal union closed families in connection with the {\it Union Closed Sets Conjecture} of Frankl. Our result may lead to potential full proof of the union closed set conjecture and several other conjectures. The part of the dissertation related to the maximum edge cuts in connected digraphs goes as follows. In a given digraph $D$, a set $F$ of edges is defined to be a {\it directed cut} if there is a nontrivial partition $(X,Y)$ of $V(D)$ such that $F$ consists of all the directed edges from $X$ to $Y$. The maximum size of a directed cut in a given digraph $D$ is denoted by $\Lambda (D)$, and we let $\mathcal{D}(1,1)$ be the set of all digraphs $D$ such that $d^{+}(v)=1$ or $d^{-}(v)=1$ for every vertex $v$ in $D$. In this part of dissertation, we prove that $\Lambda (D) \geq \frac{3}{8}(|E(D)|-1)$ for any connected digraph $D\in\mathcal{D}(1,1)$, which provides a positive answer to a problem of Lehel, Maffray, and Preissmann. Additionally, we consider triangle-free digraphs in $\mathcal{D}(1,1)$ and answer their another question.

Statistics / M. Sc. (Statistics)

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Na tentativa de facilitar o diagnÃstico dos diversos fatores que afetam a qualidade da Ãgua e antever possÃveis impactos futuros sobre o meio ambiente , sÃo adotadas aÃÃes que racionalize m o uso da Ãgua a partir da otimizaÃÃo de processos naturais ou tecnolÃgicos. A modelagem matemÃtica à um exemplo disso e, em conjunto com a Teoria Fuzzy , que permite fazer a anÃlise dos resultados sem necessidade de significativos bancos de dados, pode - se estabelecer o risco como indicador de contaminaÃÃo das Ãguas de rios, sendo de valor prÃtico na tomada de decisÃo e concessÃo de outorga de lanÃamentos. Neste estudo, foi desenvolvido um modelo matemÃtico aplicado Ãs equaÃÃes completas de Streeter - Phelps utilizando a Teoria dos nÃmeros Fuzzy , a fim de analisar o risco de contaminaÃÃo de um curso d'Ãgua que recebe agentes poluentes de mÃltiplas fontes de lanÃamento. Pelas simulaÃÃes do modelo, foram analisados diferentes cenÃrios, verificando a influÃncia d os seus parÃmetros, bem como o lanÃamento de fontes poluidoras pontuais e difusas, nos percentuais de risco. De acordo com os resultados, observou - se que a quantidade de carga lanÃada tem influÃncia no tempo de diluiÃÃo desta massa no sistema, de forma que , para maiores valores de lanÃamento, o tempo de diluiÃÃo à menor, favorecendo os processos de decaimento e formaÃÃo da camada bentÃnica; em relaÃÃo Ãs reaÃÃes fÃsicas, quÃmicas e biolÃgicas, verifica - se que os processos de sedimentaÃÃo, fotossÃntese e res piraÃÃo, para os dados mÃdios encontrados em literatura, tem pequena influÃncia no comportamento das curvas de concentraÃÃo de OD e curvas de risco, enquanto que o processo de nitrificaÃÃo tem forte influÃncia; jà a temperatura desempenha um significativo papel no comportamento do OD, onde, para valores maiores, maior serà o dÃficit OD e, em consequÃncia, aumento dos percentuais de risco. Por fim, o modelo desenvolvido como proposta de facilitar a tomada de decisÃo no controle de lanÃamento de efluentes em rios mostrou - se uma alternativa viÃvel e de valor prÃtico de anÃlise, jà que os objetivos foram alcanÃados / In an attempt to facilitate the diagnosis of the various factors that affect water quality and predict possible future impacts on the environment, actions to rationalize the use of water from the optimization of natural and technological processes are adopted. Mathematical modeling is one example and, together with Fuzzy Theory, which allows the analysis of the results without the need for significant databases, one can establish the risk as an indicator of contamination of rivers, and of practical value in decision making and allocation of grant releases. In this study, the full Streeter-Phelps equations, using the Fuzzy set Theory, was applied, in order to analyze the risk of contamination of a watercourse that receives multiple sources release pollutants. Through the model simulations, different scenarios were analyzed, and the influence of its parameters as well as the launch point and nonpoint pollution sources, in the calculation of the risk. According to the results, it was observed that the amount of discharge released influences the time of the mass dilution in the system, so that for higher values of launch, the dilution time is less favoring the formation and decay processes of benthic layer; regarding the physical, chemical and biological reactions, it appears that sedimentation processes, photosynthesis and respiration, concerning with the average data found in literature, have little influence on the behavior of the curves of DO concentration curves and risk, while the nitrification process has a strong influence; with respect to the temperature, the results showed that it plays a significant role in the behavior of DO, where, for larger values of it, the higher the DO deficit and, consequently, increase in the risk. Finally, the model developed as a proposal to facilitate the decision making in the control of discharge of effluents into rivers proved to be a viable and practical analytical alternative way, since the goals were achieved.